3D Rotation Moment Invariants

If we need rotation invariants, we work with the moment tensors as with Cartesian tensors. Each tensor product of the moment tensors, where each index is used just twice, is then the invariant to the 3D rotation. E.g.

Mii :

,

MijMij :

,

MijMjkMki :

.

The division by the suitable power of m000 is normalization to scaling.

In the following appendix, there are all linerly independent invariants that have from 1 to 8 indices in the generating tensor product. The tensor products are expressed by graphs (lists of edges), where each edge corresponds to one index and each node corresponds to one moment tensor.

If both forms are the same, we note them as square .The index j must be even.

The coefficients of the invariants are square roots of rational numbers. The conversion of the coefficients from approximate floating-point format to this format is main limitation of the increasing of the order, therefore we present here the invariants to the fifth order only. We cannot guarantee the precise conversion for higher orders.